34-O Geometry. from plan to meet boiler circumference, and carry these along horizontally to cross the vertical lines at c D ; the serpentine curve, being then traced through the numbers obtained, will represent the developed intersecting line. This may be repeated on the second half of the plate, and allowances made for flanging and welding. The boiler hole is developed by stepping-off the three distances, /$, h^ and H, with dividers, and measuring them from the vertical centre line in plan to give a, b, and c respec- tively, the remaining four segments being symmetrical. The length of plate is found by calculation. Intersections of oblique cylinder with plane, or cone with cylinder, are rarely required; but cone with plane is sometimes necessary, as in funnels for American locomotives, or conical flues such as that shewn at L, Fig. 310. The latter has been chosen as an example, and the form of plate developed at K, Fig. 315, J being the finished flue. The drawing j having been made, the outer lines are produced to meet at f, and the dotted circles struck, with gf and//as radii. Upon these are measured the circumferences at d and e respectively, and allowance made for welding and flanging. If the set-squares at hand be not long enough, the marker-off should be able to set out a right angle by the measurements of three sides of a triangle, it being easily remembered that the proportions 3, 4, 5, for base, perpendicular, and hypotenuse in turn, will serve his purpose, as can be proved by the 47th proposi- tion of Euclid's first book, thus : 32 + 42= 52 or 9 -f 16 =25 The length of arc, chord being known, is sometimes required, and may be obtained as follows:— Let *:=the half chord. r=radius of arc. a = half the angle subtended by the arc. Then - = sin a. r The angle a being found from a table of sines, Length of Arc = 2 x —- x 2w.r= '0349^.